Handling Overdispersion with Negative Binomial and Generalized Poisson Regression Models

In actuarial literature, researchers suggested various statistical procedures to estimate the parameters in claim count or frequency model. In particular, the Poisson regression model, which is also known as the Generalized Linear Model (GLM) with Poisson error structure, has been widely used in the recent years. However, it is also recognized that the count of frequency data in insurance practice often display overdispersion, i.e., a situation where the variance of the response variable exceeds the mean. Inappropriate imposition of the Poisson may underestimate the standard errors and overstate the significance of the regression parameters, and consequently, giving misleading inference about the regression parameters. This paper suggests the Negative Binomial and Generalized Poisson regression models as alternatives for handling overdispersion. If the Negative Binomial and Generalized Poisson regression models are fitted by the maximum likelihood method, the models are considered to be convenient and practical; they handle overdispersion, they allow the likelihood ratio and other standard maximum likelihood tests to be implemented, they have good properties, and they permit the fitting procedure to be carried out by using the Iterative Weighted Least Squares (IWLS) regression similar to those of the Poisson. In this paper, two types of regression model will be discussed and applied; multiplicative and additive. The multiplicative and additive regression models for Poisson, Negative Binomial and Generalized Poisson will be fitted, tested and compared on three different sets of claim frequency data; Malaysian private motor third party property damage data, ship damage incident data from McCullagh and Nelder, and data from Bailey and Simon on Canadian private automobile liability.